168
Enthymemetic Conditionals: Topoi as a guide for acceptability
Eimear Maguire
Laboratoire Linguistique Formelle (UMR 7110), Universit´e de Paris [email protected]
Abstract
To model conditionals in a way that reflects their acceptability, we must include some means of making judgements about whether antecedent and consequent are meaningfully related or not. Enthymemes are non-logical ar-guments which do not hold up by themselves, but are acceptable through their relation to a topos, an already-known general principle or pattern for reasoning. This paper uses en-thymemes and topoi as a way to model the world-knowledge behind these judgements. In doing so, it provides a reformalisation (in TTR) of enthymemes and topoi as networks rather than functions, and information state update rules for conditionals.
1 Introduction
The content of the antecedent and consequent of a conditional, not just their truth or falsity, makes a difference to whether we find the conditional ac-ceptable or not, generally rejecting those that seem disconnected (Douven,2008). If we are to model conditionals in a way that reflects their acceptabil-ity, we must include some means of making those judgements. Enthymemes are non-logical argu-ments which do not hold up by themselves, but are acceptable through their relation to a topos, an already-known general principle or pattern for reasoning. Arguments and justifications in inter-action tend to be underpinned by general princi-ples and rules of thumb, rather than being truly ‘logical’. For models of dialogue to be adequate then, these non-logical arguments need to be han-dled – namely, as proposed byBreitholtz(2014a), through incorporating enthymemes and topoi into the dialogue model. Apart from the evidence from their own acceptability conditions, which corre-late strongly with judgements of high conditional probability, conditional structures are also asso-ciated with ‘that kind of thinking’, being used
as plain-language explanations of particular topoi (e.g. “if something is a bird, then it flies” in Bre-itholtz,2014b), or used as materials on reasoning in any number of experiments (e.g.Pijnacker et al., 2009). If we are going to explicitly recognise the use of such ‘rule’ type objects in discourse, then conditionals are one place where they show up, at least sometimes.
This paper has two aims. First, to propose a for-malisation of enthymemes and topoi that is geared towards relating them to more complex rule-based world knowledge, including a distinction between knowledge about causality, non-causality, and am-biguity about causality. Second, to account for the acceptability (or not) of conditionals by proposing an enthymeme-like structure as associated with
if-conditionals, such that topoi can enhance their content and are used in judging whether a given conditional is acceptable or not. The acceptabil-ity of conditionals is linked to perceived rela-tionships between the antecedent and consequent cases: with enthymemes and topoi, whose pres-ence in the model is independently justified, we can incorporate this non-arbitrarily into the dia-logue state.
The rest of this section will provide some back-ground. Section 2 is focused on enthymemes, topoi, and specification of the alternative formal-ism, while Section 3 uses this in a proposal of update rules associated with conditionals. Lastly, Section4provides a conclusion. This paper draws on work on enthymemes and topoi elsewhere in Breitholtz (2014a,b) etc., and will likewise use Type Theory with Records (Cooper, 2012, here-after referred to by the acronym TTR) for formal-isation.
1.1 Enthymemes and Topoi
incomplete in that to be accepted, they must be identified as a specific instance of a more general pattern that is already in the agent’s resources – a topos. Topoi encode world knowledge that comes as a ‘rule of thumb’, such as characteristics typical of groups, and a speaker may hold contradictory topoi as equally valid in different scenarios, with no clash experienced unless both are used at the same time. Speakers make enthymemetic argu-ments by linking what on the surface might techni-cally be non-sequiturs, but are easily identified as an argument using accepted principles. For exam-ple, a speaker might say “Let’s go left here, it’s a shortcut”. This argument invokes the assumption that shorter routes are better, and that therefore the left turn being a shortcut is a good reason to take it – but they might equally say “it’s longer”, invok-ing an assumption that a longer route is preferable. Topoi have been proposed to be a resource available to speakers, and consequently a means to address non-monotonic reasoning (Breitholtz, 2014b), the treatment of non-logical rules as ex-pressing necessity, and contradictory claims being equally assertable, as in the route-taking example above (Breitholtz,2014a).
To these ends, they have been formalised in TTR for use in dialogue (Breitholtz and Cooper, 2011), as functions from records to record types, as in this example (Breitholtz,2014a):
(1) a. Topos:
λr :[
x :Ind
cbird : bird(x)
]([cf ly : fly(r.x)])
b. Enthymeme:
λr :[x= Tweety :Ind
cbird : bird(x)
]([cf ly : fly(Tweety)])
Both are of typeRec→RecType, and the fields of the specified record types match, but fields of the enthymeme have been restricted to specific values. A function to a record type does not by itself in-dicate what happens once we have access to that type. For these functions to be useful, they are ad-ditionally governed by a theory of action, which will license various actions that can be performed with the type, e.g. judging that the original situa-tion is addisitua-tionally of that type, judging that there exists some situation of the type described, or cre-ating something of that type (Cooper, in prep).
1.2 Conditionals
The assumption that conditionals express a propo-sition is fundamental to most linguistic work on
the topic, both that which follows the commonly accepted restrictor theory of conditional semantics based on the work ofLewis(1975),Kratzer(1986) and Heim (1982), and that which does not (e.g. Gillies(2010)).
By conditionals being ‘propositional’, we mean that adding anif-clause to some indicative clause does not fundamentally change the kind of seman-tic object it is: for indicative clause “I’m going home”, just as the conjunction ”I’m going home
and I’m watching a film” still expresses a proposi-tion, so does “If this doesn’t get interesting soon, I’m going home”.
As mentioned at the beginning, the acceptabil-ity of conditionals correlates strongly with their conditional probability: the more likely the con-sequent is in the antecedent-case, the more ac-ceptable the conditional tends to be be. Stalnaker (1970) proposed that the probability of a condi-tional and the condicondi-tional probability of the con-sequent on the antecedent are one and the same, in what is usually referred to asthe Equation. That is, the overall probabilityP(if this doesn’t get
in-teresting then I’m going home)is the same as the conditional probability P(I’m going home∣This
doesn’t get interesting). A subsequent proof by Lewis(1976) found that there is no single proposi-tion based on the antecedent and consequent such that its probability will consistently match the con-ditional probability. Therefore one could have a propositional theory of conditionals, or validate the Equation – but not both.
moder-ated by whether there appears to be a connection between antecedent and consequent (Skovgaard-Olsen et al.,2016). To make these judgements, we need to know about the relationships between the antecedent and consequent states.
As a note, this paper remains technically ag-nostic about whether the propositional or proba-bilistic analysis is correct: your mileage may vary on whether the update rules in Section 3 should also add a proposition associated with if p, q to the agent’s knowledge base, were they to be more comprehensively specified. The underlying ac-ceptability issue, and the potential use of topoi in the metrics underlying those acceptability judge-ments, means that this does not impact on the core of the proposals here.
1.3 TTR: a brief overview
Since it will be used later, this section provides a very brief introduction to TTR.
A central idea in TTR is the judgement of ob-jects as being of some type. Ifais judged to be of type T, this is written asa ∶ T. Several of these judgements, or requirements for judgements, can be collected in structured objects as records and record types. In a record type, fields consist of a label and type, while fields in a record consist of a label and a value. For a recordr to be of a record typeRT, it must have fields with the labels specified inRT, and the values in those fields inr
must be the types specified by the equivalently la-belled fields inRT. For example, the records in (2) and (3) are both of the record type (4), provided thatxis of typeT1. The type of a field need not
be stand-alone either: it may also be constructed from a predicate and arguments, like the fielddin (5).
(2) [a=x]
(3) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
a=x
b=y
c=z ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(4) [a∶T1]
(5) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣
a∶T1
d∶p(a) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
(6) ⎡⎢ ⎢ ⎢ ⎢ ⎣
a∶T1
b∶T2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (7) ⎡⎢
⎢ ⎢ ⎢ ⎣
a∶T1
c∶T3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
There also exist sub- and super-type relations between types. One record type is a subtype of another if it is a more specified version of it. This means that it has at least the same fields as the su-pertype, whose types are the same type or subtypes of the equivalent fields in the supertype. For ex-ample, (6) and (7) are different types, but are both subtypes of the more general (4). A record of type
(6) is not necessarily of type (7), but will be of type (4). Depending on whetherx∶T1,y∶T2 and
z∶T3, the record in (3) will be of all three types.
2 Enthymemes, Topoi and Other Knowledge
Given that their presence in an agent’s resources has already been motivated, topoi are a natural way to account for the required knowledge about some ‘dependence’ between antecedent and con-sequent. Enthymemes and topoi are snippets of reasoning, rather than complex networks, but they should also be related explicitly to other rule-like world knowledge, which includes the possibility of multiple relationships between more than two cases, and knowledge of explicitly causal rela-tions. If we are going to use topoi to express the kind of knowledge that also forms such networks (i.e. informative about causality or related proba-bilities), then they should be in the same form as that knowledge: the alternative, to keep rule-like topoi apart from knowledge about rule-based(ish) systems, is counter-intuitive.
Bayesian networks (a combination of directed acyclic graphs and probability distributions) are a common way to encode causal relations. They have two components, the first of which is a di-rected acyclic graph, with the various variables as nodes, and directed edges describing any direct relationships. Graphs and networks are a useful way to describe relationships, and express a more complex set of relationships than a linear chain of functions. The graph structure is in accordance with constraints about what direct parenthood in the graph can mean – that the parent is part of the minimal set of preceding nodes whose value de-termines the probability distribution of the child.
2.1 Graphical Topoi
The proposal is as follows. Topoi and enthymemes are of the same type as any other ‘relational’ knowledge, by which I mean knowledge about causal and correlational relations. This knowledge can be encoded as a graph: topoi and enthymemes as usually discussed are minimal examples, con-taining only two nodes. The direction(s) of the links between connected nodes, along with addi-tional constraints, indicate either causal or non-causal relations via directed or bi-directed links re-spectively.
Where there is a bi-directional link somewhere in a path between two nodes, their relationship is confirmed as non-causal. Where there is an ab-sence of any path between two nodes, the rela-tionship may be treated as potential independence, while where there are links in one direction only, the relationship may be treated as potential causal-ity. However, neither the potential independence or causality is locked in: there should be a dis-tinction between merely lacking information, and having confirmation about an absence. Certainty about independence or causality is expressed via constraints explicitly preventing the creation of any path that would violate them.
The choice of bi-directed rather than undirected edges to express non-causality is motivated by a desire for the difference in belief from potentially causal to non-causal to be something that changes easily (i.e. with the addition of information, not replacement of one thing with another of a differ-ent type), and for creation of a ‘casual’ (not a typo) middle-ground, where only one direction is of rel-evance and there is no strong commitment either way. It can be treated as potentially causal, being the only direction of interest, but whether this is the whole story between the two is not specified.
All this is meant to allow for a more complex set of relationships than expressed in your average topos which, as stated earlier, is a minimal case with just two nodes. The original example can be thought of as follows, graphs with only two nodes:
(8) a. Topos:
[x :Ind
cbird : bird(x)
]1 [cf ly: fly(1.x)]2 0.95
b. Enthymeme:
[
x= Tweety :Ind cbird: bird(Tweety)
]1 [cf ly: fly(Tweety)]2 0.95
Oncex is filled (as ‘Tweety’), this should be re-flected in any other nodes where the same variable appears. The confidence rating of0.95 has been somewhat arbitrarily set here for topoi to imply high confidence without certainty. Generally, the confidence rating associated with a link in a known network should be subject to change on the ba-sis of experience, increasing or decreasing as their predictions are borne out or subverted. Topoi as ‘rules of thumb’ are particularly robust to contra-dictory evidence, with the same agent in different contexts accepting and using topoi that lead to op-posite conclusions: see, for example, notions op-posites attractvs.birds of a feather flock together. Integration of ordinary learning with the potential for entrenched ‘against all evidence’ beliefs is a larger topic that is not addressed here, but will be necessary in future work.
Enthymemes are distinguished from other ar-guments by the fact they don’t hold up by them-selves, but are instead accepted on the basis of identification with a topos – this doesn’t include arguments that are accepted despite being un-supported. However, the terms enthymeme and topos will continue to be used here: this is partly for convenience, but also because once the context indicates that an enthymemetic argument is being made (such as a recognisable sugges-tion+motivation pattern like “Let’s go left here, it’s a shortcut”), an unsupported ‘enthymeme’, once accepted, can be used to establish a potential new topos (Breitholtz,2015).
2.2 Graphical Topoi in TTR
This subsection provides a treatment in TTR of the above proposal. The variable at each node is a RecT ype, representing a situation, with the probability of aRecT ypebeing across whether it is true or false (for typeT, whether ∃a ∶ T). Let
RecTypei be a RecTypeassociated with an index,
andProbInfobe a constraint on some probability. The supertype of enthymemes and topoi, rather than a function Rec→RecType, is the type
Net-work:
(9) Network=def
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
nodes :{RecTypei}
links :{⟨RecTypei, RecTypei⟩}
probs :{ProbInfo}
cindex:∀⟨x′j, yp⟩,∈links,x′j⊑rxi∈nodes,i=j, ∀⟨zq, x′′k⟩ ∈links,x′′k⊑rxi∈nodes,i=k.
clinks:∀⟨x′i, y ′
p⟩ ∈links,∃xi, yp∈nodes,x′i⊑rxi,yp′ ⊑ryp
The nodes field is the set of nodes in the graph, while the links field is the set of directed edges between them, each ‘link’ being an ordered pair. Let ⊑r indicate a subtype relation where subtyp-ing is through restriction of one or more fields i.e. not through the specification of extra fields. The first constraint cindex enforces co-indexing, that
if subtypes of a node are included in members oflinks, they all share the same index. The sec-ond constraint clinksspecifies that any members of
links are between (potentially restricted subtypes of) members ofnodes. For ease of reading and the sake of space, the constraints will not be re-peated in further examples. In a link⟨xi, xj⟩, the specification of memberxi may usej to indicate
somer ∶xj, and vice versa, e.g. whereais some field inxi andbis some field inxj, inxi we can
specify thata=j.b.
Causality, non-causal correlation and indepen-dence are interpreted on the basis of the mem-bers of links. Where a path is a sequence of in-dices ⟨1, . . . , k⟩ such that for each i, i+1 there is ⟨xi, xi+1⟩ ∈ links, the node indexediis a pre-decessor of the node indexed j (shorthand: pre-decessor(i, j,links)) if there is a path fromitoj, given the contents oflinks. In this way the setlinks
can be checked for evidence that two nodes are in a non-causal relation (if there is a bi-directional
predecessorrelation somewhere in a path between the two, e.g. if⟨xi, xj⟩,⟨xj, xi⟩ ∈links), are poten-tially independent (there is nopredecessorrelation at all between the two), or in a potentially causal relation (one is apredecessorof the other, but not the other way around). We can distinguish direct and indirect causality by whether a minimal path with a direct link⟨xi, xj⟩is possible or not. As a rule, when we talk about causality, we will mean direct causality.
Forn∶Networkcontaining nodesxi andxj, in-dependence and causality can be expressed in up-datedn′
as follows, wherea.bindicates the merge of two records, a record containing all fields from both, anda.bindicates their asymmetric merge (seeCooper and Ginzburg,2015), where in event of a field appearing in both records, the field from
bis the one found in the merge, effectively over-writing the field ofa.
(10) Independence ofiandj:
n′ =n. ⎡ ⎢ ⎢ ⎢ ⎣
cindij :¬predecessor(i, j,links)
∧¬predecessor(j, i,links) ⎤ ⎥ ⎥ ⎥ ⎦
(11) Direct causality fromitoj:
n′ =n. ⎡ ⎢ ⎢ ⎢ ⎣
ccauseij:⟨i, j⟩ ∈links
∧¬predecessor(j, i,links) ⎤ ⎥ ⎥ ⎥ ⎦
(12) Indirect causality fromitoj:
n′ =n. ⎡ ⎢ ⎢ ⎢ ⎣
cindcausij :predecessor(j, i,links)
∧¬predecessor(j, i,links) ⎤ ⎥ ⎥ ⎥ ⎦
The original example can now be rewritten as (13):
(13) Topos: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ nodes = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ [
x : Ind cbird: bird(x)
]1,[
x : Ind cf ly: fly(x)
]2 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
:{RecTypei}
links = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
⟨ [x : Ind
cbird: bird(x)
]1,[x =1.x: Ind
cf ly: fly(x)
]2⟩ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
:{⟨RecTypei, RecTypei⟩}
probs = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ P ⎛ ⎜ ⎜ ⎜ ⎝ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣
x = r.x: Ind cfly: fly(x)
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
2 ∣r : ⎡ ⎢ ⎢ ⎢ ⎢ ⎣
x : Ind cbird: bird(x)
⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 1 ⎞ ⎟ ⎟ ⎟ ⎠ = 0.95 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
:{ProbInfo} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(14) Enthymeme: as above, but all vari-ants indexed with 1 are replaced with
[x = Tweety: Ind
cbird: bird(x)
]1
AnEnthis defined as aNetworkcontaining a node that has at least one field restricted to a specific object, removing its generality. AToposis a Net-workin which no fields are restricted to a specific object.
An enthymemeemay be identified with a topos
tif its nodes and links have equivalents int, that is if for every node xi ∈ e.nodes,∃yp ∈ t.nodes such that xi ⊑ yp and for any links ⟨x′i, x′j⟩ ∈
e.links,∃⟨yp′, y′q⟩ ∈ t.links such that x′i ⊑ yp′ and
x′
j ⊑y
′
q. This may be by a clear match for the topos
fields, but may also include the types of fields in the enthymeme as subtypes of fields in the topos1.
3 Conditionals and Reasoning
Having reformalised topoi and enthymemes as an object intended for more general correlational and causal knowledge, i.e. like a Bayesian network2 we turn back to conditionals.
1
as in the example “Give a coin to the porter, he carried the bags all the way here” from (Breitholtz,2014b), where carrying someone else’s bags is recognised as a subtype of work, and the enthymemetic argument is on the basis of a topos likework should be rewarded
2
Firstly, and as mentioned at the beginning, ex-pressing this kind of relational knowledge is (both intuitively and according to empirical evidence) strongly associated with conditionals, and exis-tence of a dependence relation and high condi-tional probability usually determine their accept-ability. Van Rooij and Schulz (2019) suggest a way to combine these two features into a single measure, the relative difference the state of the parent in a relation makes to the likelihood of the child. Pleasingly, with some independence assumptions this measure works not only for the ‘causal’ direction typically expressed by condi-tionals (if there’s fire, there’s smoke), but for the reverse as expressed by evidential conditionals (if there’s smoke, there’s fire). However, for it to do so, the direction of the relationship still has to be recognised even when the ‘usual’ roles of an-tecedent as parent and consequent as child have flipped. This kind of structural knowledge is topoic.
Secondly, and while it feels almost trivial to point out, we use conditionals to tell each other new things, e.g. the speaker explaining their expe-riences with “if you done anything wrong well you get, you get the cane and anything else” (BNC,
H5G 78). When we are informed of something through the use of a conditional, we don’t nec-essarily know beforehand that they lie in such a relation: otherwise they would only be useful to draw attention to connections we haven’t made, not to tell each other things that are entirely new. Indeed,Skovgaard-Olsen et al.(2016) found evi-dence that when faced with a conditional, people assume that there is a positive connection between antecedent and consequent unless they have rea-son to believe otherwise. It is not so much that an acceptable conditional has to be backed up by pre-existing knowledge about the relation between the antecedent and consequent cases, but at the very least it should notclashwith any.
Breitholtz (2014a) mentions how an en-thymemetic argument can be recognised on the basis of the current conversational game/expected rules (with the specific example of knowledge that a suggestion may be followed by the speaker pro-viding a motivation), or by an explicit lexical cue. With the above in mind, I will suggest that use of anif-conditional is one such linguistic cue.
3.1 Enthymemetic Conditionals
The overall suggestion is as follows. If -conditionals are associated with the making of enthymeme-like arguments. Note that I say “enthymeme-likearguments”, not “enthymemetic arguments”. Enthymemes depend on identifica-tion with a previously-known topos, while condi-tionals can be used to teach new relations, rather than just make statements that rely on existing knowledge to make sense. Although they are structured like the characterisation of enthymemes and topoi above, they are not all strictly speak-ing ‘enthymemetic’. The content of a conditional can be checked against the topoi in the agent’s re-sources. Given a match with a topos, an enhanced version can be added to the agent’s knowledge.
Even without a guiding topos, conditionals al-low us to express or learn information via an as-sumption that there is a positive connection be-tween antecedent and consequent – provided we do not already know that the two are independent, or that the consequent shouldn’t follow from the antecedent. If no supporting topos is found, a more minimal version can be added without the benefit of any extra details a topos might have pro-vided.
The direction of antecedent as parent is ‘default’ in the sense that it should be preferred if distinct topoi in both directions are available, and is the di-rection assumed in case neither a supporting topos nor a conflicting one is found. The topoi in an agent’s resources may conflict with each other, and by necessity one of them was learned first: despite this, a conditional does not lead to formation of an acceptable enthymeme when such a clashing topos is already present. If there only exists a potential match for the nodes in a topos that specifies there is definitely no link, or is a conflicting link, then the conditional should be rejected.
The processes of comparing a potential enthye-meme with a topos and of updating structured knowledge on the basis of a conditional can be thought of algorithmically as follows:
(15) Match between an enthymeme and topos:
Search known topoi for topos with a node matching the first enthymeme node
If none: no match,false.
If found: check topos for nodes matching each further node in enthymeme.
If found: check each edge in enthymeme has an equivalent in topos.
If any failure: resume searching topoi. If found: check any constraints in enthymeme have an equivalent in topos.
If found: match,true.
If any failure: resume searching topoi.
(16) Enhancing an enthymeme with a topos:
Make new copy of topos.
For each node in topos with an equivalent in enthymeme, add any further specification. For any node in topos with no equivalent node in enthymeme, but with elements also found in a node that was further specified, update ac-cordingly.
(17) Updating with a conditional:
Check for conflicting topos. If found,reject.
If not found, check for topos matching
ant→consequivalent link.
If found,enhanceant→consandadd. If not found, check for topos matching
ant←consequivalent link.
If found,enhanceant←consandadd. If not found,addant→cons.
Below are illustrations of what should be un-derstood from the evidential conditional “If the glass fell, the cat pushed it”, given knowledge of a topos equivalent toif someone pushes something, the thing falls.
(18) Ant. content:
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
x= obj3
cglass= glass(obj3)
cfall= fall(obj3)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(19) Cons. content:
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
x= obj4
y= obj3
ccat= cat(obj4)
cpush= push(obj4, obj3)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(20) Topos:
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
x: Ind
y: Ind
cpush: push(x, y) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
1 [
x=1.y: Ind
cfall: fall(x) ]2 0.95
(21) Enhanced enthymeme:
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
x=obj4: Ind
y=obj3: Ind
ccat: cat(x)
cpush: push(x, y) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
1
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
x=obj3: Ind
cglass: glass(x)
cfall: fall(x) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
2 0.95
The following subsections describe dialogue state update rules associated with conditionals, characterised in TTR.
3.2 Enthymemetic Conditionals in TTR
To begin with, the type of an information state is minimally given as (22), broadly following the de-cisions for the place of enthymemes and topoi in Breitholtz(2014a) etc.
(22) InfoState=def
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
priv :[Topoi :{Topos}]
shrd :
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
enths :{Enth}
Topoi :{Topos}
Moves : list(LocProp)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(23) Update rule=def
[pre : InfoState
effects : Infostate]
The information state has two parts: the agent’s private resources, and their representation of the shared context. Theprivateresources include a set of general topoi which they can use as resources. A publicTopoifield tracks which topoi have been introduced onto the dialogue gameboard. The gen-eral form for update rules is given in (23):pre de-scribes the preconditions for states to which the rule can be applied, andeffectsthe changes.
Next we will add a few useful functions on the basis of some of the content of Section 2.1: a means to describe whether there is a successful match between an enthymeme and a topos, and a means to reference the result of an enthymeme that has been enriched by the content of a topos.
(24) enthMatch(e : Enth, t : Topos) : Bool, trueiff all of the following hold
(i) Alle’s nodes are subtypes oft’s nodes: ∀xi∈e.nodes,∃yp∈t.nodes
such thatxi⊑yp,
(ii) Alle’s links are subtypes oft’s links: ∀⟨x′i, x′j⟩ ∈e.links,∃⟨yp′, y′q⟩ ∈t.links such thatx′
i⊑y
′
pandx′j⊑y
′ q,
(iii) For any constraints on links in e, the same constraints hold for equivalent links int:
∀cindij ∈e,∃cindpq∈tor cindqp ∈t,
xi ∈ e.nodes, yp ∈t.nodes,xi ⊑ yp and
xj ∈e.nodes, yq∈t.nodes,xj⊑yq. Likewise for all ccauseij ∈e, there is an
equivalent ccausepq ∈t.
(25) enhanceEnth(e: Enth, t: Topos) : Enth,
e′such thate′is an asymmetric merge oft
where the sets in nodes, links and probs
undergo asymmetric union such that for any nodes xi ∈ e.nodes, yp ∈ t.nodes,
xi ⊑ yp, the corresponding node zu ∈
e′
.nodes=yp .xi.
Likewise for any subtypesx′
i andy′p,x′i⊑
y′
p in members of e.links, t.links, e.probs
andt.probs.
That is, the asymmetric aspect of the merge is at the level of the indexed nodes, not the fields containing them.
The update rules for each case are given in the subsections below. There are three rules given: where there is a supporting topos in the ‘default’ direction, where there is not but there is a support-ing topos in the reverse direction, and where there is neither support nor a clash.
3.2.1 Recognising a supporting topos
First are the update rules for when the agent has a topos linking the two parts of the conditional: The update in case of a supporting topos in the
ant→consdirection is given in (26):
(26) default direction, ant→cons:
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ pre : ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ priv : ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
Topoi :{Topos}
t : Topos
cmember :t∈Topoi
cdef :enthMatch(x∶X,t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
shrd :[Moves[0] = Assert(if(a, b)) : LocProp] ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ effects : ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ shrd : ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
enths = pre.shrd.enths
∪enhanceEnth(x∶X,t) :{Enth}
Topoi = pre.shrd.Topoi∪pre.priv.t:{Topos} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
whereXis the type
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
nodes ={a.sit-type1, b.sit-type2}:RecTypei
links ={⟨a.sit-type1, b.sit-type2⟩}:⟨RecTypei,RecTypei⟩
probs ={P(b.sit-type2∣r : a.sit-type1)=0.95}: ProbInfo ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
This rule may be applied following assertion of a conditional, where an agent knows a topos tthat matches an enthymeme based on the content of the conditional, with a link from antecedent to con-sequent. In this case, the agent may add such an enthymeme enhanced with the topos to theirenths, and add the underlying topos to the set of currently active topoi in the conversation.
Where such an option does not exist, a topos with only a link from consequent to antecedent can be used, as described in (27). The enthymeme added toenthsin this case will contain a link only in theant←consdirection.
(27) alternative direction, ant←cons:
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ pre : ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
shrd :[Moves[0] = Assert(if(a, b)) : LocProp]
priv : ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
Topoi :{Topos}
t : Topos
cmember :t∈Topoi
cno-def :∄t
′
, t′
∈Topoi∧enthMatch(x∶X,t′)
calt :enthMatch(y∶Y,t)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ effects : ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ shrd : ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
enths = pre.shrd.enths
∪enhanceEnth(y∶Y,t):{Enth}
Topoi = pre.shrd.Topoi∪pre.priv.t:{Topos} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
whereXis as defined in (26), andY is the type
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
nodes ={a.sit-type1, b.sit-type2}:RecTypei
links ={⟨b.sit-type2, a.sit-type1⟩}:{⟨RecTypei,RecTypei⟩}
probs ={P(a.sit-type1∣r : b.sit-type2)=0.95}:{ProbInfo} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Relative to (26), the update rule for this case has a constraint in its preconditions that there are no topoi with a link in the ant→cons direction, and the enthymeme is instead enhanced by a topos that does support the alternative order.
3.2.2 New information
The last rule describes the case where the agent’s known topoi have neither evidence about a link be-tween the antecedent or consequent, the definite absence of one, or a conflicting one. In this case, an ‘enthymeme’ with a link in the ant→cons di-rection may be added to enthssolely on the ba-sis of the conditional content. No additional topos is added to the list of active topoi – the process for generalising an acceptable enthymeme to a re-usable topos is not addressed here.
The shorthand for presence of a clashing topos is given in (28) as enthClash. An enthymeme clashes with a topos where the equivalent parent nodes lead to mutually exclusive child nodes, i.e. child nodes where a true type cannot be formed from their meet.
(28) enthClash(e: Enth,t: Topos) : Bool, trueiff
∃⟨p′i, q′j⟩ ∈t.links, p′i⊑pi, q′j⊑qj, and¬T, whereT=yj′.q′j
(29) neither support nor opposing knowledge:
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
pre :
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
shrd :[Moves[0] = Assert(if(a, b)) : LocProp]
priv :
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
Topoi :{Topos}
cno-clash:∄t, t∈Topoi∧enthClash(x:X,t)
cno-def:∄t, t∈Topoi∧enthMatch(x:X,t)
cno-alt:∄t, t∈Topoi∧enthMatch(y:Y,t)
cno-ind:∄t, t∈Topoi∧enthMatch(z∶Z,t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
effects :[shrd :[enths = pre.shrd.enths∪x∶X]] ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
whereX,Y are as in (26), (27), andZis the type
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
nodes ={a.sit-type1, b.sit-type2}:RecTypei
links =∅:{⟨RecTypei,RecTypei⟩}
probs =∅:{ProbInfo}
cind12:¬predecessor(1,2,links)
∧¬predecessor(2,1,links) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Relative to the previous two update rules, the preconditions in this rule specify that there is no known topos that supports an enthymeme with a link between the antecedent and consequent in ei-ther direction, which has an explicit constraint en-forcing independence between the two, or which otherwise clashes with the possible conditional en-thymeme.
4 Conclusion
The acceptability of a conditional is often deter-mined by the conditional probability of the con-sequent on the antecedent, and recognition of some meaningful link between the two. However, both intuitively and according to experimental ev-idence, positive acceptability judgements can still be made without fore-knowledge of such a con-nection. This paper presented two proposals on the basis that the knowledge enabling these judge-ments is topoic, integrating these factors into the representation of the dialogue state and agent re-sources. First, a formalisation of enthymemes and topoi as graphs was presented, on the grounds that they should be in the same form as other knowl-edge about causal and correlational relationships. Second, update rules for conditionals using topoi and enthymemes were presented, drawing on topoi to recognise the presence and direction of a ‘mean-ingful’ connection between antecedent and conse-quent, and making an assumption of one in the ab-sence of any evidence.
There are several avenues for further work. Most work focuses on declarative conditionals, the most common form by far. However, conditional clauses are also used to form conditionalised ques-tions and directives. The proposals here should be related to these forms, whether because to an ex-tent they apply in those cases too, or because this topoic association is exclusive to declarative con-ditionals. This paper has also said nothing about more standard propositional aspects of condition-als. The proposals here about structural knowl-edge associated with conditionals should be inte-grated with this more standard fare.
Acknowledgements
Thanks to Jonathan Ginzburg for helpful discussion of and feedback on this work, and to the reviewers
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